Inquests can currently only be held for children who showed signs of life after being born.
The proposals would allow coroners to investigate the deaths of babies, after 37 weeks of pregnancy.
Christian Labour MP Sharon Hodgson's third child was born stillborn.
She told Premier the current law doesn't make sense.
"If there's a sign of life after they've been born and the baby still dies, they get an inquest but they can be showing signs of life during the birth right up to being born and just don't make it over the line," she said during Premier's News Hour.
"Then there's no inquest that takes place."
Nine babies are stillborn every day in the UK.
Health minister Jackie Doyle-Price said holding independent inquests would help bereaved parents get answers over what went wrong following the death of their child.
The new powers would allow coroners to investigate and make recommendations to help prevent future avoidable deaths.
Doyle-Price said: "We want to do everything we can to make pregnancy safer, by continually learning to improve the care on offer so fewer people have to experience the terrible tragedy of losing a child and those who do get the answers and support they deserve.
"Rates of stillbirths in England are the lowest on record, but we're committed to delivering on our ambition in the NHS Long-Term Plan to accelerate action to halve this number by 2025.
"This is a complex issue and it's important we get it right by listening carefully to those who are affected by these issues, so I urge everybody to have their say on this consultation.
"By sharing your experiences, you can ensure any decision we make puts women, loved ones and their babies first."
However, Sue Monckton-Rickett, chair of the Association of Christian Counsellors told Premier the added level of scrutiny could potentially worsen parent's grief.
She explained: "For some people that added legal process - the prolonged process could actually mean that they find that their grief is somehow put on hold or it's just one traumatic event that they don't want to have to face."
The joint consultation from the Ministry of Justice and Department for Health and Social Care will close on 18th June.
Listen to Premier's Eno Adeogun speaking with Labour MP Sharon Hodgson:
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